Multilevel Modeling for Psychology and the Social Sciences: Models, Graphs, and PracticeHsiu-Ting Yu 游琇婷 Hsiu-Ting Yu游琇婷
中文
Web edition · 2026

Multilevel Modeling

for Psychology and the Social Sciences

Models, Graphs, and Practice

Hsiu-Ting Yu游琇婷
Department of Psychology, National Chengchi University, Taipei, Taiwan

Each level of a multilevel structure has its own units and its own variation. In the three-level empty model of Chapter 8, pupil scores vary around their class mean, class means around their school mean, and school means around the grand mean.

Pupils within classes within schools Three stacked planes for the three levels of a multilevel structure: schools at level 3, classes at level 2, and pupils at level 1. Each school is linked to its classes and each class to its pupils, and the units of one school share a color. LEVEL 3 schools k β00k ~ 𝒩(γ000, φ02) LEVEL 2 classes j π0jk ~ 𝒩(β00k, τ02) LEVEL 1 pupils i yijk ~ 𝒩(π0jk, σ2)
Pupils within classes within schools, one plane per level. The units of one school share a color; point to a school to follow its classes and pupils.
01

About this book

Data in psychology, education, and the social sciences are almost always grouped. Pupils are taught in classes within schools, patients are treated by therapists within clinics, employees work in teams within firms, and repeated measurements are nested within persons. Multilevel models represent the variation at every level at once, relate it to explanatory variables measured at the corresponding level, and give inferences that respect the dependence the grouping creates.

This book develops multilevel modeling from nested data and the random intercept model to growth models, generalized linear mixed models, cross-classified and multiple-membership structures, location-scale models, multilevel structural equation and measurement models, and causal inference with clustered data. Every model is written three ways, as level-specific equations, as a single mixed model, and as a marginal model for the mean and covariance of the outcome, and every model is drawn as a Unified Model Graph before it is fitted.

R carries the worked examples throughout; each chapter closes with a software crosswalk to SAS, SPSS, Stata, Mplus, HLM, and JASP, a chapter summary, key terms, five kinds of exercises, and a pointer to a self-paced lab. Derivations are kept in Technical Detail boxes and in Appendix B, so that the main text can be read at the level of a first graduate course.

AudienceGraduate students in psychology, education, sociology, and related fields after a regression course; advanced undergraduates with a strong regression background; doctoral students in quantitative methods; applied researchers who need a reference for a specific extension.
Background assumedMultiple regression with interactions and dummy coding, ANOVA, basic probability, and a willingness to read matrix notation (Appendix A is a refresher).
GlossaryAppendix F lists 548 key terms with their Traditional Chinese equivalents as used in Taiwan (中英術語對照).
02

Draw before you fit: the UMG lexicon

Unified Model Graphs encode each part of a model by one visual channel. Fill shows whether a variable is observed, shape shows its support, the diamond marks a fixed parameter, and plates show replication, so that nested plates are the levels of the hierarchy. The full reference is in Appendix D.

Observed variableShaded circle: a measured, continuous random variable.
Latent variableOpen circle: an unobserved random variable, such as a random effect.
Fixed parameterDiamond: an unknown constant to be estimated, such as a fixed effect.
Categorical variableSquare: shaded if observed (an item response), open if latent (a class).
PlateRounded box: exchangeable replication; nested plates are the levels of the hierarchy.
EdgesSingle-headed arrow: regression or loading. Double-headed arrow: a covariance; drawn as a loop on one node, a variance.
Five Multilevel Data Structures Drawn as Unit Diagrams
Figure 1.1. Five Multilevel Data Structures Drawn as Unit Diagrams
The Empty, Random Intercept, and Random Intercept and Slope Models as Unified Model Graphs
Figure 2.4. The Empty, Random Intercept, and Random Intercept and Slope Models as Unified Model Graphs
Where Variables Live: Three Data Structures Drawn With Plates
Figure 1.2. Where Variables Live: Three Data Structures Drawn With Plates
03

Reading paths

Each core chapter corresponds to roughly one three-hour lecture, and every chapter restates its notation, so the book can be read in several orders.

One-semester graduate course

Chapters 1–18, 19, 20, 23

The course sequence, with Chapters 13, 16, and 17 read as introductions; fourteen content weeks.

Advanced undergraduate course

Chapters 1–7, 10–12, 14, 19

The core models, inference, model building, and growth, with the Technical Detail boxes omitted.

Second-semester advanced course

Chapters 13, 16, 17, 21, 22, 24–32

Bayesian estimation, design and missing data, intensive longitudinal data, and the extensions of Parts VI and VII.

Reference use

Appendix E and any single chapter

The software crosswalk and troubleshooting guide, together with any chapter, each of which restates its notation.

04

Contents

Thirty-two chapters in seven parts, followed by seven appendices. The table of contents is also available in the left panel on every page.

Chapters 5–32 are password protected, each with its own password. Chapters 1–4 and Appendices A–F are open to all readers. Enter a chapter’s password here or on the chapter’s page to open that chapter on this device. Appendix G (solutions to the exercises) also has its own password and will be opened to all readers at a later date.
I

Foundations

  1. 1Multilevel Data, Multilevel Questions
  2. 2Three Views of One Model: Notation, Equations, and Graphs
  3. 3From ANOVA and Regression to Multilevel Models
  4. 4Exploring and Managing Multilevel Data
II

The Core Models

  1. 5The Random Intercept Model
  2. 6Random Slopes, Variance Functions, and Cross-Level Interactions
  3. 7Centering, Contextual Effects, and the Fixed-Versus-Random Debate
  4. 8Three-Level and Higher-Order Hierarchies
III

Estimation and Inference

  1. 9Estimation: Likelihood, REML, and Algorithms
  2. 10Inference for Fixed Effects
  3. 11Inference for Variance Components and Global Fit
  4. 12Predicting Random Effects: Empirical Bayes, BLUP, and Shrinkage
  5. 13Bayesian Multilevel Modeling
IV

Building, Checking, Designing, Reporting

  1. 14Model Building Strategies
  2. 15Assumptions, Diagnostics, and Heterogeneous Variance
  3. 16Sample Size, Power, and the Design of Multilevel Studies
  4. 17Missing Data and Complex Survey Designs
  5. 18Interpreting, Reporting, and Reproducing Multilevel Analyses
V

Longitudinal and Intensive Data

  1. 19Growth Models: Longitudinal Data as Multilevel Data
  2. 20Covariance Structures and Serial Correlation
  3. 21Within-Person Dynamics and Intensive Longitudinal Data
  4. 22Growth Models Across Frameworks: Multilevel and Latent Curve Models
VI

Generalized and Extended Models

  1. 23Multilevel Models for Binary Outcomes
  2. 24Ordinal, Nominal, and Count Outcomes
  3. 25Multilevel Survival and Event-History Models
  4. 26Cross-Classified and Multiple-Membership Models
  5. 27Multivariate Outcomes, Dyadic Data, and Multilevel Meta-Analysis
  6. 28Location-Scale Models: Variability as an Outcome
VII

Latent Variables, Causality, Frontiers

  1. 29Multilevel Structural Equation Modeling, Mediation, and Moderation
  2. 30Multilevel Measurement: Factor Models and Explanatory Item Response Models
  3. 31Causal Inference With Clustered Data
  4. 32Frontiers and Alternatives: Cluster-Robust, GEE, Fixed-Effects, Regularized, and Tree-Based Mixed Models
A–G

Appendices

  1. AMatrix Algebra and Distribution Refresher
  2. BLikelihood, REML, and Empirical Bayes Derivations
  3. CData Set Catalog and Codebooks
  4. DUMG Notation Reference and Figure Style Guide
  5. ESoftware Crosswalk: One Model in Every Program, and Troubleshooting
  6. FGlossary of Terms in English and Traditional Chinese (中英術語對照)
  7. GSolutions to Selected Exercises
05

In every chapter

Every chapter follows the same template, so recurring elements are marked the same way throughout the book.

Learning objectivesWhat a reader should be able to do after the chapter.
The model, three waysHierarchical, linear mixed, and marginal forms side by side.
Key ideaThe one claim a paragraph or section exists to establish.
Technical detailDerivations that can be skipped on a first reading.
PitfallCommon errors of specification, estimation, and interpretation.
Reviewer asksThe questions a critical reviewer would put to the analysis.
Taiwan caseApplications with data and questions from Taiwan.
Software crosswalkThe chapter’s models in SAS, SPSS, Stata, Mplus, HLM, and JASP.
Chapter summaryThe main results of the chapter in a few statements.
Exercises and labConcept checks, analytical, computational, simulation, and applied exercises, and a lab.
06

Copyright and use

© 2026 Hsiu-Ting Yu (游琇婷). All rights reserved.

This web edition is provided for personal study and teaching. You are welcome to link to any page and to cite the book as shown under “Cite this book”; reproduction, redistribution, or adaptation of the text, figures, tables, or solutions requires the author’s written permission.

R code examples are released under the MIT License. Data sets remain subject to the terms of their original providers (see Appendix C).